How to use these statistics practice questions
These statistics practice questions are designed as a short mixed set rather than a list to read passively. For each question, choose an answer before opening the explanation in the practice screen. Then identify the exact step that decided the result: a definition, a formula, an assumption or an interpretation.
The set covers descriptive statistics, probability, sampling distributions, confidence intervals, hypothesis testing, correlation and regression. The numbers are small enough to calculate by hand, but the explanations use the language expected in a written statistics answer.
1. Conditional probability
A screening test is positive in 90% of people who have a disease and in 10% of people who do not. The disease affects 1% of the population. What is the probability that a person with a positive result has the disease?
A disease has prevalence 1%. A test has sensitivity 90% and a false-positive rate of 10%. Given a positive result, what is the probability that the person has the disease?
The probability is P(D|+) = (0.90 × 0.01) / [(0.90 × 0.01) + (0.10 × 0.99)] = 0.009 / 0.108 = 0.0833, or 8.3%. The strongest distractor is 90%, which is the sensitivity: P(+|D), not the requested P(D|+).
The important distinction is between the probability of a positive test when disease is present and the probability of disease when the test is positive. Bayes’ theorem changes the conditioning direction, and the low prevalence makes false positives important.
2. Expected value and variance
A discrete random variable takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. Find its expected value.
A random variable X has P(X=0)=0.2, P(X=1)=0.5 and P(X=2)=0.3. What is E(X)?
E(X) = ΣxP(X=x) = (0×0.2) + (1×0.5) + (2×0.3) = 1.1. The strongest distractor is 1.0, which can result from treating the three listed values as equally likely rather than using their stated probabilities.
For an expected value, multiply each possible value by its probability and add the products. Do not divide by the number of possible values unless the distribution is uniform.
3. Standardising a normal observation
A measurement is normally distributed with mean 70 and standard deviation 8. What is the z-score for an observation of 86?
A variable has mean 70 and standard deviation 8. What is the z-score for an observation of 86?
The z-score is z = (x − μ)/σ = (86 − 70)/8 = 16/8 = 2. The strongest distractor is 16, which is the raw difference from the mean and has not been divided by the standard deviation.
A z-score states how many standard deviations an observation lies from the mean. Keeping the subtraction and division in that order prevents the common error of reporting the raw distance instead.
4. Sampling distributions
A population has standard deviation 12. A random sample of 36 observations is taken. What is the standard error of the sample mean?
A population has standard deviation 12, and a random sample has size 36. What is the standard error of the sample mean?
The standard error is σ/√n = 12/√36 = 12/6 = 2. The strongest distractor is 6, which is √n; it is the divisor, not the standard error itself.
The standard deviation describes individual observations; the standard error describes the variability of sample means. Increasing the sample size reduces the standard error by the square root of the sample size, not by the sample size itself.
5. Confidence intervals
A sample mean is 52, the population standard deviation is known to be 10, and n = 100. Find the 95% confidence interval using z = 1.96.
A sample has mean 52, known population standard deviation 10 and size 100. Using z=1.96, what is the 95% confidence interval for the population mean?
The standard error is 10/√100 = 1. The margin of error is 1.96×1 = 1.96, so the interval is 52 ± 1.96 = (50.04, 53.96). The strongest distractor is (51.80, 52.20), which uses 0.2 as the standard error rather than 1.
Show the standard error and margin of error separately. A confidence interval is an interval-producing procedure: in repeated samples, intervals constructed this way would contain the fixed population mean at the stated long-run rate. It is not correct to say that there is a 95% probability that this already calculated fixed mean lies inside this particular interval.
6. Null hypotheses and p-values
A test produces p = 0.03 at significance level 0.05. What is the appropriate conclusion?
A statistical test gives p=0.03. The significance level is 0.05. Which conclusion is appropriate?
Because 0.03 is below 0.05, reject the null hypothesis at the 5% significance level. The strongest distractor is the claim that the null hypothesis has a 3% probability of being true: a p-value is calculated under the null hypothesis and is not that posterior probability.
A good conclusion states the decision, the threshold and the direction of evidence without overstating it. Statistical significance does not establish practical importance, causation or the probability that a hypothesis is true.
7. Type I and Type II errors
A researcher rejects a true null hypothesis. Which error has occurred?
A researcher rejects the null hypothesis even though it is true. What type of error is this?
Rejecting a true null hypothesis is a Type I error, with probability controlled by the chosen significance level in the relevant testing framework. The strongest distractor is Type II error, which is failing to reject a false null hypothesis.
Keep the two errors as a paired definition. Type I means a false positive decision; Type II means a false negative decision. Bias and sampling error are different problems and should not be used as substitutes for these definitions.
8. Correlation and causation
A study finds r = -0.82 between temperature and heating-fuel use across several months. What does this statistic describe?
A study reports r=-0.82 between outdoor temperature and heating-fuel use. Which interpretation is best?
r=-0.82 indicates a strong negative linear association: higher temperature tends to occur with lower fuel use in these data. The strongest distractor is the causal claim; correlation alone does not rule out confounding or establish causation.
The sign gives the direction and the magnitude gives the strength of the linear association. Correlation is unit-free, while a regression slope has units and depends on the scales of both variables.
9. Simple linear regression
A fitted model is ŷ = 4 + 2.5x. What is the predicted value when x = 6?
A fitted regression model is ŷ=4+2.5x. What value does it predict when x=6?
Substitute x=6: ŷ = 4 + (2.5×6) = 4 + 15 = 19. The strongest distractor is 15, which is the contribution from the slope but omits the intercept of 4.
Write the intercept first, then substitute the predictor value. The predicted value is not necessarily an observed value, and extrapolation beyond the range of the data requires particular caution even when the arithmetic is correct.
10. Choosing a test
Two independent groups have quantitative outcomes that are approximately normally distributed, with similar variances. Which test compares their means?
Two independent groups have quantitative outcomes that are approximately normal, with similar variances. Which test is most appropriate for comparing their means?
An independent two-sample t-test compares the means of two independent groups under the stated assumptions. The strongest distractor is the paired t-test, which is for matched observations or repeated measurements on the same units.
Test selection depends on the outcome type, number of groups, independence and distributional assumptions. State those features before naming a test; this is more reliable than memorising a test name in isolation.
What to revise after the set
Do not record only your total score. Classify each error:
- Definition error: for example, confusing a p-value with the probability that the null hypothesis is true.
- Formula error: for example, using σ/n instead of σ/√n for a standard error.
- Arithmetic error: the correct method with an incorrect calculation.
- Interpretation error: a numerically correct result expressed as causation or practical importance without evidence.
- Test-selection error: choosing a paired test for independent groups, or using a quantitative-outcome method for categorical data.
A compact revision board should contain the formulas and distinctions that the questions exposed, not every line of working. MySummaries can turn that board into a spaced deck and track which ideas repeatedly cost you marks.
Use the weakest section to choose your next five questions. If the same error appears twice, turn it into a one-sentence rule followed by one worked example.
You lost this mark twice: what is the difference between P(+|disease) and P(disease|+), and which one is sensitivity?
The aim is not to complete more questions without review. It is to make the next attempt discriminate between similar ideas: conditional probability versus sensitivity, standard deviation versus standard error, and statistical significance versus practical importance.
How MySummaries helps
Build a statistics board from your own notes, then use it to generate flashcards, written practice papers and marked explanations. The platform can keep recurring errors in a remediation tray so your next session starts with the concepts that need another attempt. Open MySummaries.